Sun Yuan Hsieh

Fault-free Hamiltonian cycles in faulty arrangement graphs

The arrangement graph A/sub n,k/, which is a generalization of the star graph (n-k=1), presents more flexibility than the star graph in adjusting the major design parameters: number of nodes, degree, and diameter. Previously, the arrangement graph has proved Hamiltonian. In this paper, we further show that the arrangement graph remains Hamiltonian even if it is faulty. Let |F/sub e/| and |F/sub v/| denote the numbers of edge faults and vertex faults, respectively. We show that A/sub n,k/ is Hamiltonian when 1) (k=2 and n-k/spl ges/4, or k/spl ges/3 and n-k/spl ges/4+[k/2]), and |F/sub e/|/spl les/k(n-k)-2, or 2) k/spl ges/2, n-k/spl ges/2+[k/2], and |F/sub e/|/spl les/k(n-k-3)-1, or 3) k/spl ges/2, n-k/spl ges/3, and |F/sub e/|/spl les/k, or 4) n-k/spl ges/3 and |F/sub v/|/spl les/n-3, or 5) n-k/spl ges/3 and |F/sub v/|+|F/sub e/|/spl les/k. Besides, for A/sub n,k/ with n-k=2, we construct a cycle of length at least 1) [n!/(n-k!)]-2 if |F/sub e/|/spl les/k-1, or 2) [n!/(n-k)!]-|F/sub v/|-2(k-1) if |F/sub v/|/spl les/k-1, or 3) [n!/(n-k)!]-|F/sub v/|-2(k-1) if |F/sub e/|+|F/sub v/|/spl les/k-1, where [n!/(n-k)!] is the number of nodes in A/sub n,k/.

Hamiltonian-laceability of star graphs

Suppose that G is a bipartite graph with its partite sets of equal size. G is said to be strongly Hamiltonian-laceable if there is a Hamiltonian path between every two vertices that belong to different partite sets and there is a path of (maximal) length N - 2 between every two vertices that belong to the same partite set, where N is the order of G. In other words, a strongly Hamiltonian-laceable graph has a longest path between every two of its vertices. In this paper, we show that the star graphs with dimension four or larger are strongly Hamiltonian-laceable.

Conditional Diagnosability of Augmented Cubes under the PMC Model

Processor fault diagnosis has played an important role in measuring the reliability of a multiprocessor system, and the diagnosability of many well-known multiprocessor systems has been widely investigated. The conditional diagnosability is a novel measure of diagnosability by adding an additional condition that any faulty set cannot contain all the neighbors of any node in a system. In this paper, we evaluate the conditional diagnosability for augmented cubes under the PMC model. We show that the conditional diagnosability of an n-dimensional augmented cube is 8n - 27 for n≥5.

Constructing edge-disjoint spanning trees in locally twisted cubes

The use of edge-disjoint spanning trees for data broadcasting and scattering problem in networks provides a number of advantages, including the increase of bandwidth and fault-tolerance. In this paper, we present an algorithm for constructing n edge-disjoint spanning trees in an n-dimensional locally twisted cube. Since the n-dimensional locally twisted cube is regular with the common degree n, the number of constructed trees is optimal.

Pancyclicity on Möbius cubes with maximal edge faults

A graph G = (V, E) is said to be pancyclic if it contains cycles of all lengths from 4 to |V| in G. Let Fe be the set of faulty edges. In this paper, we show that an n-dimensional Mobius cube, n ≥ 1, contains a fault-free Hamiltonian path when |Fe| ≤ n - 1. We also show that an n- dimensional Mobius cube, n ≥ 2, is pancyclic when |Fe| ≤ n - 2. Since an n-dimensional Mobius cube is regular of degree n, both results are optimal in the worst case.

Hamiltonian path embedding and pancyclicity on the Mobius cube with faulty nodes and faulty edges

A graph G=(V, E) is said to be pancyclic if it contains fault-free cycles of all lengths from 4 to |V| in G. Let F/sub v/ and F/sub e/ be the sets of faulty nodes and faulty edges of an n-dimensional Mobius cube MQ/sub n/, respectively, and let F=F/sub v//spl cup/F/sub e/. A faulty graph is pancyclic if it contains fault-free cycles of all lengths from 4 to |V-F/sub v/|. In this paper, we show that MQ/sub n/-F contains a fault-free Hamiltonian path when |F|/spl les/n-1 and n/spl ges/1. We also show that MQ/sub n/-F is pancyclic when |F|/spl les/n-2 and n/spl ges/2. Since MQ/sub n/ is regular of degree n, both results are optimal in the worst case.

Dynamic Programming on Distance-Hereditary Graphs

In this paper, we define a one-vertex-extension tree for a distance-hereditary graph and show how to build it. We then give a unified approach to designing efficient dynamic programming algorithms for distance-hereditary graphs based upon the one-vertex-extension tree, We give linear time algorithms for the weighted vertex cover and weighted independent domination problems and give an O(n2) time algorithm to compute a minimum fill-in and the treewidth for a distance-hereditary graph.

Longest fault-free paths in star graphs with edge faults

In this paper, we aim to embed longest fault-free paths in an n-dimensional star graph with edge faults. When n/spl ges/6 and there are n-3 edge faults, a longest fault-free path can be embedded between two arbitrary distinct vertices, exclusive of two exceptions in which at most two vertices are excluded. Since the star graph is regular of degree n-1, n-3 (edge faults) is maximal in the worst case. When n/spl ges/6 and there are n-4 edge faults, a longest fault-free path can be embedded between two arbitrary distinct vertices. The situation of n<6 is also discussed.

Strongly Diagnosable Product Networks Under the Comparison Diagnosis Model

The notion of diagnosability has long played an important role in measuring the reliability of multiprocessor systems. Such a system is t-diagnosable if all faulty nodes can be identified without replacement when the number of faults does not exceed t, where t is some positive integer. Furthermore, a system is strongly i-diagnosable if it can achieve (t + 1)-diagnosability, except for the case where a nodes neighbors are all faulty. In this paper, we investigate the strong diagnosability of a class of product networks, under the comparison diagnosis model. Based on our results, we can determine the strong diagnosability of several widely used multiprocessor systems, such as hypercubes, mesh-connected k-ary n-cubes, torus-connected k-ary n-cubes, and hyper-Petersen networks.

Conditional Edge-Fault Hamiltonicity of Matching Composition Networks

A graph G is called Hamiltonian if there is a Hamiltonian cycle in G. The conditional edge-fault Hamiltonicity of a Hamiltonian graph G is the largest k such that after removing k faulty edges from G, provided that each node is incident to at least two fault-free edges, the resulting graph contains a Hamiltonian cycle. In this paper, we sketch common properties of a class of networks, called matching composition networks (MCNs), such that the conditional edge-fault hamiltonicity of MCNs can be determined from the found properties. We then apply our technical theorems to determine conditional edge-fault hamiltonicities of several multiprocessor systems, including n-dimensional crossed cubes, n-dimensional twisted cubes, n-dimensional locally twisted cubes, n-dimensional generalized twisted cubes, and n-dimensional hyper Petersen networks. Moreover, we also demonstrate that our technical theorems can be applied to network construction.